USING SPARSE POLYNOMIAL CHAOS EXPANSIONS FOR

THE GLOBAL SENSITIVITY ANALYSIS OF GROUNDWATER

LIFETIME EXPECTANCY IN A MULTI-LAYERED

HYDROGEOLOGICAL MODEL

G. Deman, K. Konakli, B. Sudret, J. Kerrou, P. Perrochet, H. Benabderrahmane

CHAIR OF RISK, SAFETY AND UNCERTAINTY QUANTIFICATION

STEFANO-FRANSCINI-PLATZ 5CH-8093 ZURICH

Risk, Safety &Uncertainty Quantification

Data Sheet

ETH Project: Sensitivity analysis in computational hydrogeology

Project ID: 34971

Journal: -

Report Ref.: RSUQ-2015-001

Arxiv Ref.: http://arxiv.org/abs/1502.01477 - [stat.CO]

DOI: -

Date submitted: January 20th, 2015

Date accepted: -

Using sparse polynomial chaos expansions forthe global sensitivity analysis of groundwater

lifetime expectancy in a multi-layeredhydrogeological model

G. Deman1, K. Konakli∗2, B. Sudret2, J. Kerrou1, P.Perrochet1, and H. Benabderrahmane3

1The Centre for Hydrogeology & Geothermics (CHYN), University of Neuchatel,Rue Emile Argand 11, CH-2000 Neuchatel, Switzerland

2ETH Zurich, Institute of Structural Engineering, Chair of Risk, Safety &Uncertainty Quantification, Stefano-Franscini-Platz 5, CH-8093 Zurich,

Switzerland3Andra, 1-7 rue Jean Monnet, 92298 Chatenay-Malabry Cedex, France

Abstract

The present study makes use of polynomial chaos expansions tocompute Sobol’ indices in the frame of a global sensitivity analysis ofhydro-dispersive parameters in a synthetic multi-layered hydrogeolog-ical model. The model introduced in this paper is a simplified verticalcross-section of a segment of the subsurface of the Paris Basin. This15-layer numerical model is aimed at exploring the behavior of ground-water and solute fluxes when accounting for uncertain advective anddispersive parameters. Applying conservative ranges, the uncertaintyin the input variables is propagated upon the mean lifetime expectancyof water molecules departing from a specific location within a highlyconfining layer situated in the middle of the numerical model. Thesensitivity analysis indicates that the variability in the mean lifetimeexpectancy can be sufficiently explained by the uncertainty in thepetrofacies, i.e. the sets of porosity and hydraulic conductivity, ofonly a few layers of the model.

∗Corresponding author, e-mail: konakli@ibk.baug.ethz.ch

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1 Introduction

With the improvement of computing power, numerical modeling has becomea popular tool for understanding and predicting various kinds of subsurfaceprocesses addressed in the fields of geology and hydrogeology. However, theincomplete/imprecise knowledge of the underground system frequently com-pels the modeller to make a number of approximations and assumptions withregard to the geometry of geological structures, the presence of discontinu-ities and/or the spatial distribution of hydro-dispersive parameters in theirmodels Renard (2007). These uncertainties can possibly lead to large vari-abilities in the predictive modeling of subsurface processes and thus, it be-comes of major importance to account for the aforementioned assumptionsin the frame of uncertainty and sensitivity analyses. Uncertainty analysis(UA) aims at quantifying the variability of a given response of interest as afunction of uncertain input factors, whereas sensitivity analysis (SA) has thepurpose to identify the input factors responsible for this variability. Hence,SA determines the key variables to be described in further detail in order toreduce the uncertainty on the predictions of a model.

Methods of SA are typically classified in two categories: local SA andglobal SA methods. The former investigate effects of variations of the inputfactors in the vicinity of nominal values, whereas the latter aim at quanti-fying the output uncertainty due to variations in the input factors in theirentire domain. Among several global SA methods proposed in the literature,of interest herein is SA with Sobol’ sensitivity indices, which belongs to thebroader class of variance-based methods Saltelli et al. (2008). These methodsrely upon the decomposition of the response variance as a sum of contribu-tions of each input factor or combinations thereof; unlike regression-basedmethods, they do not assume any kind of linearity or monotonicity of themodel.

Various methods have been investigated for computing the Sobol’ indicesthat were first defined in Sobol’ (1993), see e.g. Archer et al. (1997); Sobol’(2001); Saltelli (2002); Sobol’ and Kucherenko (2005); Saltelli et al. (2010).In these papers, Monte Carlo simulation is used as a tool to estimate thesesensitivity indices. This has revealed extremely costly, although more efficientestimators have been recently proposed Sobol et al. (2007); Janon et al.(2013). In the last few years, new approaches using surrogate models havebeen introduced in the field of global sensitivity analysis Storlie et al. (2009);Zuniga et al. (2013). A popular method to compute the Sobol’ indices,originally introduced by Sudret (2008), is by post-processing the coefficientsof the polynomial chaos expansion (PCE) of the response quantity of interest.PCE constitute an efficient UA method in which the key concept is to expand

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the model response onto a basis made of orthogonal polynomials in the inputvariables. Once a PCE representation is available, the Sobol’ indices can becalculated analytically with elementary operations at almost no additionalcomputational cost. Sparse PCE make the approach even more efficient, asshown in Blatman and Sudret (2010a).

In the frame of the stochastic modeling of subsurface flow and mass trans-port, PCE meta-models have proven to be robust and comprehensive tools inperforming SA at low computational cost. As an example, applying a PCE-based global SA upon a fine-grid numerical model of flow and mass transportin a heterogeneous porous medium, Fajraoui et al. (2011) and Younes et al.(2013) established the transient effect of uncertain flow boundary conditions,hydraulic conductivities and dispersivities on solute concentrations at givenobservation points. Sochala and Le Maıtre (2013) propagated uncertain soilparameters upon three different physical models of subsurface unsaturatedflow. Their study proved the higher efficiency of PCE meta-models, in com-parison to a classical Monte-Carlo method, for representing the variability ofthe output quantity at low computational cost. In the frame of radionuclidetransport simulation in aquifers, Ciriello et al. (2013) analyzed the statisticalmoments of the peak solute concentration measured at a specific locationas a function of the conductivity field, the dispersivity coefficients and thepartition coefficients associated to the heterogeneous media. The compari-son of the Sobol’ indices obtained for various degree of PCE meta-modelsshowed that low-degree PCE models can yield reliable indices while consid-erably reducing the computational burden. Formaggia et al. (2013) usedPCE-based sensitivity indices to investigate effects of uncertainty in hydro-geological variables on the evolution of a basin-scale sedimentation process.However, the various aforementioned contributions consider simplified mod-els for the description of subsurface flow and mass transport. A more realisticrepresentation of these processes is employed in the present study by usinga large-scale simulator.

In the scope of the deep geological storage of radioactive wastes, ANDRA(French National Radioactive Waste Management Agency) has conductedmany studies to assess the potentiality of a clay-rich layer for establishinga mid to long-lived radioactive waste disposal in the subsurface of the ParisBasin. The thick impermeable layer from Callovo-Oxfordian (COX) age hasbeen extensively studied (Delay et al., 2006; Distinguin and Lavanchy, 2007;Enssle et al., 2011) together with the two major limestone aquifers, in place ofthe Dogger and the Oxfordian sequences (Brigaud et al., 2010; Linard et al.,2011; Landrein et al., 2013), encompassing the claystone formation. A recentstudy (Deman et al., 2015) used a high-resolution integrated Meuse/Haute-Marne hydrogeological model (AND, 2012) to compute the average time for

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water molecules departing from a given area in the COX to reach the limitsof the numerical model. SA over hydro-dispersive parameters in 14 hydro-geological layers proved that the Dogger and Oxfordian limestone sequenceshave a large influence on the residence time of groundwater. Indeed, ad-vection processes occurring in permeable layers strongly influence the watertransit in the subsurface of the Paris Basin, in contrast to the slow-motiondiffusive processes taking place in impermeable rocks.

However, the analysis of the effect of uncertainties related to other advective-dispersive parameters, such as boundary conditions, orientations and anisotropiesof hydraulic conductivity tensors or magnitudes of dispersion parameters,represents a great effort that cannot be carried out with the integrated modelat reasonable computational costs. Addressing the issue of performing UAwith the use of high-resolution numerical models of geological reservoirs,Castellini and co-workers (Castellini et al., 2003) established that numericalmodels built at the coarse scale, but covering a reasonable number of geolog-ical and geostatistical features, can be particularly informative in capturingthe main subsurface processes at low computational costs.

The present study introduces a vertical two-dimensional multi-layeredhydrogeological model representing a simplification of the underground mediaof the Paris Basin in the vicinity of the site of Bure and does not integratethe complex geometry of the layers, neither does it include the numerousdiscontinuities or heterogeneities observed in the field. It must be emphasizedthat the use of this model is focused on numerical issues, sensitivity analysisand calibration purposes and the results cannot be considered with respectto the real situation.

The main objective of the present work is to assess the effect of multipleadvective-dispersive parameter on the mean lifetime expectancy (MLE) ofwater molecules departing from a target zone in the central layer. The MLEcorresponds to the average time required for a given solute, taken at a specificlocation, to reach any outlet of the model. Conservative uncertainty rangesare defined for the input factors analyzed in the frame of a SA relying onthe estimation of PCE-based Sobol’ indices. This study provides a prelim-inary assessment on the relative effect of factors governing the MLE in thesubsurface of the Paris Basin and recommendations are made for the applica-tion on the high-resolution integrated Meuse/Haute-Marne hydrogeologicalnumerical model of the Paris Basin.

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2 The numerical model

2.1 Geometry and finite element mesh

Originally inspired by the COUPLEX numerical model from Bourgeat et al.(2004), the present model stands as a vertical two-dimensional (x-z) cross-section of 25,000 × 1,040 meters representing a segment of the Paris Basinsubsurface. The mesh is discretized into 5 × 5 meters square elements fora total of 1,040,000 elements. In order to subdivide the domain into en-tities related to geological formations, the main features of the subsurfacewere extracted from the lithostratigraphic log of the deep EST433 borehole(Landrein et al., 2013) in the vicinity of the experimental site of Bure (Haute-Marne, France). Therefore, the model consists of 15 hydrogeological layerscharacterized by tabular geometries, uniform thicknesses and homogeneousparameters. Figure 1 summarizes the geometry of the model and gives anoverview on the succession and thicknesses of layers.

The bottom layer stands as a 110 m thick low-permeability layer at-tributed to the Toarcian marl formation (T). Overlying the latter, the suc-cession of carbonate formations from the Dogger sequence is subdivided into5 layers of which the total thickness attains 250 m in the numerical model.The sequence encompasses the Bajocian (D1) and Bathonian limestones (D3and D4) representing the main aquifer formations of the Dogger, a clasticdominated interval (”Marnes de Longwy”, D2) separating the two. The Dog-ger sequence is topped with a thin oolithic limestone from Lower Callovian(”Dalle Nacree”, C1), implemented as a 15 m thick layer in the model. Thelatter marks the transition with the thick, highly impermeable, claystone for-mation of Callovo-Oxfordian age (C2) of which the thickness reaches 150 min the model. In the numerical simulations, a target zone (TZ) located in themiddle of layer C2 (Figure 1) represents the location for the computation ofthe output quantity of interest.

The low-permeability COX layer is overlaid by a limestone sequence ofthe Oxfordian age. The latter is incorporated as a 260 m thick formation sub-divided into 6 hydrogeological entities. A relatively confining layer from theUpper Argovian (C3ab) rests directly on the COX and is followed by perme-able formations of the Rauracian-Sequanian sequence (L1a to L2c). A thickinterval of marls and argillaceous limestones from Kimmeridgian age (K1-K2) covers the whole and is implemented as a 160 m thick low-permeabilitylayer. The top layer is a 120 m thick confining formation attributed to theTithonian (K3). The latter outcrops in the vicinity of Bure.

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Figure 1: Geometry and geological layers with the localization of the targetzone (vertical exaggeration: 20).

2.2 Governing equations and model outputs

In the numerical simulations the flow is governed by the steady-state equation

∇ · q = 0, (1)

where q = −K ∇H, is the Darcian flux vector [L T−1], K is the tensor of hy-draulic conductivity [L T−1] and H is the hydraulic head [L]. The anisotropyAK in the components of the tensor of hydraulic conductivity is defined asthe ratio between the hydraulic conductivities in the two principal directions:AK = Kz/Kx.

Here, it is assumed that K has orthotropic properties. Considering ahydraulic conductivity tensor Kp of which the components are mapped intothe Cartesian system and given along their principal direction, Xp, the tensorK in the global Cartesian space is retrieved by means of the rotation matrixR with the expression

K = RT Kp R. (2)

For the two-dimensional problem considered, the rotation matrix R is defined

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in terms of the Euler angle θ (in degree) as

R =

(cos θ sin θ− sin θ cos θ

). (3)

In the present study, steady-state flow simulations are carried out to-gether with the computation of the lifetime expectancy probability densityfunction (PDF) at any point x in the domain. Under stationary conditions(i.e. steady-state flow), the lifetime expectancy PDF addresses the proba-bility distribution of the time required for a solute, taken at any positionx, to leave the domain. In its formulation, the lifetime expectancy PDFassimilates the forward advective-diffusive transport equation (ADE) to theFokker-Planck (forward Kolmogorov) equation measuring the random motionof solute particles (Uffink, 1989). For more details on the computation of thelifetime expectancy PDF the reader is referred to Cornaton and Perrochet(2006a,b) and Kazemi et al. (2006).

Based on the ADE, the lifetime expectancy PDF is computed using thebackward transport equation requiring reversed flow directions (q := -q) aswell as adjusted downstream boundary conditions. The lifetime expectancyPDF gE(x, t) at any point x in the domain is then governed by

φ∂gE∂t

= ∇ · (q gE + D ∇gE), (4)

where φ is the effective porosity [-] and where D is the dispersion tensor

φD = (αL − αT )q⊗ q

||q|| + αT ||q|| I + φ Dm I, (5)

where I is the identity matrix, Dm is the coefficient of molecular diffu-sion [L2 T−1], αL and αT are the longitudinal and transverse componentsof the macro-dispersion tensor [L] respectively. In the present study, theanisotropy in the macro-dispersion tensor is determined with the coefficientAα = αT/αL.

The straightforward computation of the first moment of the lifetime ex-pectancy PDF is the so-called mean lifetime expectancy (MLE) E(x) [T] atany position x, governed by

−∇ · (q E + D ∇E) = φ, (6)

where it can be seen that the porosity φ [-] acts as the sink term in the agingprocess.

The target zone (TZ) comprises a set of 1,947 nodes in layer C2, coveredby a rectangle which lateral and vertical extensions are x= [18,440 ; 21,680], z

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= [425 ; 435] (Figure 1). In the present study, the arithmetic mean of E(x, z)calculated at each of these 1,947 nodes stands for the output response ofinterest and is used in the subsequent analysis. It can be seen as the averagetime for a solute originating from the TZ to reach any outlet of the model.

The finite element simulator GroundWater (Cornaton, 2007) was em-ployed to solve Eq. (1)-(6) using the finite element technique. A single runof steady-state flow and MLE computation takes about 120 seconds using aparallel solver with 6 CPU.

The reader should note that the use of a 2D vertical model to solve forthe hydro-dispersive processes cannot capture correctly the real behavior ofthe Paris Basin subsurface because, apart from being a simplified model, itomits the lateral flow and dispersion along the third dimension. This has theeffect to underestimate the magnitude of the modeled processes Kerrou andRenard (2010). It is however recognized that this bias is equivalent for allthe layers considered, thus the interpretation of the SA results obtained withthe 2D cross-section may be generalized to a synthetic 3D case employingthe same settings.

2.3 Flow boundary conditions

The fully saturated model considers stationary flow conditions in a confinedaquifer which are implemented as Dirichlet type flow boundary conditions.These flow BCs are imposed on nodes located on top of the numerical modelas well as on both lateral limits of the two limestone sequences (Figure 2).

Regional piezometric maps based on field measures (Linard et al., 2011)were used to constrain the hydraulic gradients in both carbonate sequences.The flow BCs imposed on the lateral boundaries of the two limestone se-quences derive from a 25 km transect starting from the Gondrecourt troughand extending in a North-West direction, the main regional flow direction.The hydraulic gradient set on top of the model corresponds to the averagetopographic gradient of the region covered by the transect.

Under these conditions, the general groundwater flow direction is ori-ented from right to left. The proportions of the total outflowing rates areapproximately 2%, 60% and 38% for the top of the model, the Oxfordian andthe Dogger discharge boundaries, respectively. In layer C2, the groundwaterflows downward in the very right part of the model and then upward in theremainder, with a hydraulic gradient inversion in the vicinity of the TZ (seeFigure 2). As a summary, the flow BCs are gathered in Table 1.

To account for uncertainties in the flow BCs, the hydraulic gradients inthe two limestone sequences and on the top of the model are considered asuncertain input factors included in the following SA (Section 4). A change

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Table 1: Flow boundary conditions.

Boundary Position Hydraulic headright Oxfordian x = 25000, z = [500, 760] H = 305 mleft Oxfordian x = 0, z = [500, 760] H = 230 mright Dogger x = 25000, z = [110, 360] H = 295 mleft Dogger x = 0, z = [110, 360] H = 275 mtop of the model x = [0, 25000], z = 1040 H = 225 + 85x/25000elsewhere no flow

in the hydraulic gradients may shift the position of the vertical groundwaterflux inversion in layer C2, and thus the MLE calculated at the TZ.

Figure 2: Flow boundary conditions and head contours (vertical exaggera-tion: 20).

2.4 Hydraulic conductivity and porosity values

Many studies have undertaken the inventory of hydraulic conductivity (K)and porosity (φ) values in the various geological formations of the Paris Basin.

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For a large number of wells and boreholes within a wide area around theexperimental site of Bure, laboratory and field measurements were conductedto provide {K,φ} datasets for the two limestone sequences (Brigaud et al.,2010; Linard et al., 2011; Fourre et al., 2011; Delay and Distinguin, 2004;Delay et al., 2007a).

However, very few {K,φ} datasets are available for the four low-permeabilityformations implemented in the present model (i.e. K3, K1-K2, C2 and T).Hence, data extracted from the literature (Cosenza et al., 2002; Delay et al.,2007b, 2006; Enssle et al., 2011; Mazurek et al., 2011; Vinsot et al., 2011),and employed in previous studies (Contoux et al., 2013; de Hoyos et al., 2012;Goncalves et al., 2004a,b), were used to define the uncertainty ranges for the{K,φ} sets in these layers.

In the geological formations of the Oxfordian and Dogger sequences, largevariabilities of the {K,φ} couples are noticed with the presence of dependen-cies (e.g. low K and low φ values are correlated). However, to simplify theconceptual approach in a first stage, a perfect dependence between log10(K)and φ is defined in each layer by making use of mathematical functions ap-proximating the relationship between these two. In the sequel, both param-eters are referred to as a whole under the name petrofacies. This approachreduces the computational burden of the subsequent SA by avoiding the useof correlation functions between the two uncertain factors. For each layer,the estimated value of hydraulic conductivity K is retrieved through a rela-tionship: log10(K) = f(φ).

Although no explicit information is available on the following, the geo-logical formations are believed to feature anisotropic hydraulic conductivitytensors K, i.e. anisotropy in the two principal components of the tensor(Kx and Kz) defined as the ratio AK = Kz/Kx. The hydraulic conductivityvalues deriving from the {K,φ} distributions in each layer are attributed tothe longitudinal component of the hydraulic conductivity tensor, Kx. In thenominal case, AK = 0.1 is assumed for every layer of the model.

Preferential flow directions are supposedly taking place within each indi-vidual layer. For each layer, the Euler angle θ defines the orientation of thehydraulic conductivity tensor Kp in the Cartesian space (see Eq. (2)-(3)). Inthe nominal case, θ = 0 degree is assumed in every layer, which correspondsto the two principal components of the hydraulic conductivity tensors Kp

being oriented along the x and z axes. The orientation of the groundwaterflux q in the model is principally due to the static hydraulic gradients ∇Hresulting from flow BCs implemented on the edges of the model. Note how-ever that the Euler angle θ may locally change the orientation of q in a givenlayer and thus drive the groundwater into adjacent layers where magnitudesmight be different. This phenomenon may have a significant effect on the

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MLE calculated from the target zone, which is explored in Section 4.Table 2 summarizes the nominal values for φ and the corresponding Kx in

each of the 15 hydrogeological layers comprised in the model. The values forφ correspond to the mean value (or the median value of the CDF) of the dis-

tribution in each layer, whereas the values for Kx derive from approximationfunctions. As a reminder, the present study assumes homogeneous parame-ters in every layer. Although this feature is unrealistic, it is recalled that thepurpose of this preliminary study is to bring insights into the global effectof equivalent advective-dispersive parameters of the multi-layered hydrogeo-logical model and to provide recommendations for a similar application on ahigh-resolution integrated hydrogeological model of the Paris Basin.

Table 2: Nominal values for the porosity (φ) and the estimated longitudinal

hydraulic conductivity (Kx) in the 15 hydrogeological layers.

Layer Kx [m/s] φ [-]K3 9.01E−09 0.0100

K1-K2 4.53E−09 0.1150L2c 1.10E−06 0.1389L2b 3.46E−07 0.1110L2a 1.62E−07 0.1139L1b 1.49E−05 0.1604L1a 1.17E−06 0.1549

C3ab 4.59E−08 0.0984C2 1.99E−13 0.1580C1 1.89E−06 0.0470D4 1.65E−05 0.0905D3 1.76E−06 0.1016D2 2.62E−07 0.0623D1 3.23E−06 0.0688T 1.95E−12 0.0810

2.5 Dispersion parameters

The mean lifetime expectancy formulation (Eq. (6)) is an advective-dispersivesolute transport equation, where the longitudinal and transverse componentsof the macro-dispersion tensor (αL and αT respectively) control the particlesdispersion. These two uncertain factors depend particularly on the rocktype, on the tortuosity of the porous media and also on the scale considered.Homogeneous values of αL and αT are set within each layer, with the values

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αL = 15 m and Aα = αT/αL = 0.1 considered in the entire numerical modelin the nominal case.

As mentioned previously, no decay or adsorption effects are accountedin the computation of the MLE. The coefficient of molecular diffusion cor-responds to the theoretical self-diffusion coefficient for the water molecule,Dm = 2.3 10−9 m2/s.

3 Polynomial chaos expansions for sensitivity

analysis

Let us denote by M the computational model describing the behavior ofthe considered physical system. Let X = {X1, . . . , XM} denote the M -dimensional random input vector with joint PDF fX(x) and marginal PDFsfXi

(xi), i = 1, . . . ,M . Due to the input uncertainties represented by X,the quantity of interest becomes random. The computational model is thusconsidered as the map

X ∈ DX ⊂ RM 7−→ Y =M(X) ∈ R, (7)

where DX is the support of X. In the description of the theoretical frame-work hereafter, we assume that the components of X are independent, whichis the case for the model in the present study.

As explained in the Introduction, the aim of global sensitivity analysis isto identity random input variables and combinations thereof with significantcontributions to the variability of Y , as described by its variance. A concisedescription of the employed method of PCE-based Sobol’ sensitivity indices isgiven in the following; for further details on the topic, the reader is referred toSudret (2008) and Blatman and Sudret (2010a). The extension to the caseof mutually dependent random variables is presented in Kucherenko et al.(2012); Li et al. (2010).

3.1 Sobol’ indices

Assuming that the function M(X) is square-integrable with respect to theprobability measure associated with fX(x), the Sobol’ decomposition of Y =M(X) in summands of increasing dimension is given by Sobol’ (1993)

M(X) =M0 +M∑

i=1

Mi(Xi) +∑

1≤i≤j≤MMij(Xi, Xj) + . . .+M12...M(X) (8)

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or equivalently, by

M(X) =M0 +∑

u 6=∅Mu(Xu), (9)

whereM0 is the mean value of Y , u = {i1, . . . , is} ⊂ {1, . . . ,M} are indexsets and Xu denotes a subvector of X containing only those componentsof which the indices belong to u. The number of summands in the aboveequations is 2M − 1.

The Sobol’ decomposition is unique under the condition∫

DXk

Mu(xu)fXk(xk)dxk = 0, if k ∈ u, (10)

where DXkand fXk

(xk) respectively denote the support and marginal PDFof Xk. Eq. (10) leads to the orthogonality property

E [Mu(Xu)Mv(Xv)] = 0, if u 6= v. (11)

The uniqueness and orthogonality properties allow decomposition of the vari-ance D of Y as

D = Var [M(X)] =∑

u6=∅Du, (12)

where Du denotes the partial variance

Du = Var [Mu(Xu)] = E[M2

u(Xu)]. (13)

The Sobol’ index Su is defined as

Su = Du/D, (14)

and describes the amount of the total variance that is due to the uncer-tainties in the set of input parameters Xu. By definition,

∑u6=∅ Su = 1.

First-order indices, S(1)i , describe the influence of each parameter Xi consid-

ered separately, also called main effects. Second-order indices, S(2)ij , describe

influences from pairs of parameters {Xi, Xj}. Higher-order indices describecombined influences from larger sets of parameters.

The total sensitivity indices, SToti , represent the total effect of an inputparameter Xi, accounting for its main effect and all interactions with otherparameters. They are derived from the sum of all partial sensitivity indicesSu that involve parameter Xi, i.e.

SToti =∑

IiDu/D, Ii = {u ⊃ i}. (15)

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It follows that SToti = 1 − S∼i, where S∼i is the sum of all Su with u notincluding i.

Evaluation of the Sobol’ indices by Monte Carlo simulation is based ona recursive relationship which requires computing 2M Monte Carlo integralsinvolving M(X). Clearly, this is not affordable when the computationalmodel is a time-consuming algorithmic sequence. On the other hand, whenPCE of the quantity of interest are available, Sobol’ indices can be obtainedanalytically at almost no additional computational cost.

3.2 Polynomial chaos expansions

3.2.1 Computation of polynomial chaos expansions

A PCE approximation of Y = M(X) in Eq. (8) has the form Xiu andKarniadakis (2002)

Y =MPCE(X) =∑

α∈AyαΨα(X), (16)

where {Ψα,α ∈ A} is a set of multivariate polynomials that are orthonormalwith respect to fX , with multi-indices α = (α1, . . . , αM), and yα denotesthe corresponding polynomial coefficients.

The multivariate polynomials that comprise the PCE basis are obtainedby tensorization of appropriate univariate polynomials, i.e.

Ψα(X) =M∏

i=1

ψ(i)αi

(Xi), (17)

where ψ(i)αi (Xi) is a polynomial of degree αi in the i-th input variable belonging

to a family of polynomials that are orthonormal with respect to fXi. For

standard distributions, the associated family of orthonormal polynomials iswell-known (e.g. a uniform variable with support [−1, 1] is associated withthe family of Legendre polynomials), whereas a general case can be treatedthrough an isoprobabilistic transform of X to a basic random vector. Theset of multi-indices A in Eq. (16) is determined by an appropriate truncationscheme. In the present study, a hyperbolic truncation scheme is employed,which corresponds to selecting all multi-indices satisfying

‖ α ‖q=(

M∑

i=1

αqi

)1/q

≤ p. (18)

for appropriate 0 < q ≤ 1 and p ∈ N Blatman and Sudret (2010b).

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Once the basis has been specified, the set of coefficients y = {yα, α ∈ A}may be computed by minimizing the mean-square error of the approximationover a set of realizations of the input vector, E = {x(1), . . . ,x(N)}, calledexperimental design. Efficient solution schemes are obtained by consideringthe regularized problem

y = arg minυ∈RcardA

N∑

i=1

(M(x(i))−

∑

α∈AυαΨα(x(i))

)2

+ C‖υ‖21, (19)

where ‖υ‖1 =∑cardA

j=1 |υj| and C is a non-negative constant. A nice featureof the above regularized problem is that it provides a sparse meta-modelby disregarding insignificant terms from the set of predictors. In the presentapplication, we solve Eq. (19) using the hybrid Least Angle Regression (LAR)method as originally proposed in Blatman and Sudret (2011). Hybrid LARemploys the LAR algorithm Efron et al. (2004) to select the best set ofpredictors and subsequently, estimates the coefficients with standard least-squares minimization.

3.2.2 Error estimates

A good measure of the accuracy of PCE is the mean-square error of the

residual, ErrG = E[(Y − Y

)2], called generalization error. In practice, this

could be estimated by Monte Carlo simulation using a sufficiently large setof realizations of the input vector, Xval = {x1, . . . ,xnval

}, called validationset. The estimate of the generalization error is given by

ErrG =1

nval

nval∑

i=1

(M(xi)−

∑

α∈AyαΨα(xi)

)2

. (20)

The relative generalization error, errG, is estimated by normalizing ErrGwith the empirical variance of Yval = {M(x1), . . . ,M(xnval

)}.However, PCE are typically used as surrogate models in cases when evalu-

ating a large number of model responses is not affordable. It is thus desirableto get an error estimate of ErrG using only the information obtained from theexperimental design. One such error measure is the Leave-One-Out (LOO)error Allen (1971). The idea of the LOO cross-validation is to set apart onepoint of the experimental design, say x(i), and use the remaining points tobuild the PCE, denotedMPCE\i . The LOO error is obtained after alternatingover all points of the experimental design, i.e.

ErrLOO =1

N

N∑

i=1

(M(x(i))−MPCE\i(x(i)))2. (21)

15

Although the above definition outlines a computationally demanding pro-cedure, algebraic manipulations allow evaluation of the LOO error from asingle PCE based on the full experimental design. Let us denote by hi the i-th diagonal term of matrix Ψ(ΨTΨ)−1ΨT, where Ψ = {Ψij = Ψj(x

(i)), i =1, . . . , N ; j = 1, . . . , cardA}. Then, the LOO error can be computed as

ErrLOO =1

N

N∑

i=1

(M(x(i))−MPCE(x(i))

1− hi

). (22)

The relative LOO error, errLOO, is obtained by normalizing ErrLOO withthe empirical variance of Y = {M(x(i)), . . . ,M(x(N))}. Because this errorestimate may be too optimistic, we subsequently employ a corrected estimate,given by Chapelle et al. (2002)

err∗LOO = errLOO

(1− cardA

N

)−1 (1 + tr((ΨTΨ)−1)

). (23)

This corrected LOO error is a good compromise between fair error estimationand affordable computational cost.

3.3 Sobol’ indices from polynomical chaos expansions

Let us consider the PCE Y = MPCE(X) of the quantity of interest Y =

M(X). It is straightforward to obtain the Sobol’ decomposition of Y in ananalytical form by observing that the summands MPCE

u (Xu) in Eq. (9) canbe written as

MPCEu (Xu) =

∑

α∈Au

yαΨα, (24)

where Au denotes the set of multi-indices that depend only on u, i.e.

Au = {α ∈ A : αk 6= 0 if and only if k ∈ u}. (25)

Clearly,⋃Au = A. Consequently, due to the orthogonality of the PCE

basis, the partial variance Du reduces to

Du = Var[MPCE

u (Xu)]

=∑

α∈Au

yα2, (26)

whereas the total variance reads

D = Var[MPCE(X)

]=∑

α∈Ayα

2. (27)

16

Accordingly, the Sobol’ indices of any order can be obtained by a mere com-bination of the squares of the PCE coefficients. For instance, the first-orderSobol’ indices are given by

S(1)i =

∑

α∈Ai

yα2/D, Ai = {α ∈ A : αi > 0, αi 6=j = 0}, (28)

whereas the total Sobol’ indices are given by

SToti =∑

α∈AToti

yα2/D, AToti = {α ∈ A : αi > 0}. (29)

It is evident that once a PCE representation of Y =M(X) is available,the complete list of Sobol’ indices can be obtained at a nearly costless post-processing of the PCE coefficients requiring only elementary mathematicaloperations.

4 Results and discussion

Figure 3 provides an overview of the distribution of the mean lifetime ex-pectancy (MLE) throughout the entire model in the nominal case. In thiscase, the parameters are distributed homogeneously in each of the 15 layers;Kx and φ take on the values given in Table 2 for each layer, whereas for alllayers the anisotropy ratio is AK = 0.1, the Euler angle is θ = 0 degree, thelongitudinal component of the macro-dispersion tensor is αL = 15m and theanisotropy ratio is Aα = 0.1. The hydraulic gradients follow the boundaryconditions settings described in Section 2.3.

Because of its highly confining properties, the middle layer (C2) presentsvalues of MLE > 40,000 years. On average, it takes approximately 75,000years for a solute departing from the target zone (TZ) to reach any outlet ofthe model. Much lower MLE values are found in the two aquifer sequences,with the Oxfordian displaying slightly smaller values. The effect of conduc-tive layers is clearly distinguishable as fringes of low MLE values stretch inlayers D4, L1a and L1b in particular. As a result of the low permeability inthe top two layers (K3 and K1-K2) and the bottom layer (T), water moleculescan take more than a 100,000 years to flow through the model.

In the following, we compute the PCE-based Sobol’ indices for the MLEat the TZ by implementing the theory presented in Section 3. We conduct theanalysis in two stages: we begin with a simplified description of the input byaccounting for the uncertainty in the petrofacies only (case 1); in the sequel,we consider a higher-dimensional random input encompassing the entirety

17

Figure 3: Spatial distribution of mean lifetime expectancy in the referencecase (vertical exaggeration: 10).

of hydro-dispersive parameters described in Section 2 as well as the flow BC(case 2).

Note that the term porosity, φ, is construed in the discussion of Sobol’ in-dices. Since the values for the hydraulic conductivities are retrieved throughapproximation functions, the estimation of the sensitivity for the φ variablesis implicitly associated to that of the Kx variables in the respective layers.This is singularly important when interpreting the Sobol’ indices for aquiferformations where the hydraulic conductivity governs the ageing process (seeSection 4.3).

Computations of the PCE and Sobol’ indices are performed with theuncertainty quantification software UQLab (Marelli and Sudret (2014)).

4.1 Case 1: 15 input random variables

In the first case, the uncertain input comprises the petrofacies P in each ofthe 15 layers of the hydrogeological model. As stated before, a deterministicrelationship log10(Kx) = f(φ) is assumed for each layer, i.e. the uncertaintyregarding the petrofacies P of a layer is treated through the porosity φ, result-ing in a random input vector of dimension M = 15. The uncertain porositiesare modeled as independent uniform random variables, each bounded by thevalues φ(min) and φ(max) listed in Table 3 together with the respective coeffi-cients of variation (CoV); the porosity ranges along the model cross-sectionare shown graphically in Figure 4. The bounds φ(min) and φ(max) representthe 1st and 9th deciles of the corresponding CDF derived from porosity valuesmeasured in each geological layer. This approach is justified by the presence

18

of local extreme measures that cannot be representative for the whole layer.Bounds for the Kx parameters are also provided in Table 3, consistently withthe log10(Kx) = f(φ) approximation functions, and presented graphically inFigure 5.

Table 3: Ranges of porosity φ with the respective CoV and the estimatedpermability values K in the 15 geological layers.

Layer φ(min) [-] φ(max) [-] CoV K(min)x [m/s] K

(max)x [m/s]

K3 0.0840 0.1160 0.0924 3.3734e-10 2.4078e-07K1-K2 0.0870 0.1430 0.1406 9.8116e-11 2.0928e-07

L2c 0.1019 0.1759 0.1538 3.6186e-08 2.6212e-06L2b 0.0645 0.1574 0.2417 8.7318e-10 6.3950e-06L2a 0.0651 0.1627 0.2474 4.7005e-10 9.9336e-06L1b 0.1375 0.1833 0.0824 3.4324e-09 2.8913e-04L1a 0.0991 0.2107 0.2080 3.1165e-08 2.1523e-06

C3ab 0.0747 0.1221 0.1391 7.8488e-09 1.2945e-06C2 0.1284 0.1876 0.1082 5.0349e-14 6.2570e-13C1 0.0142 0.0799 0.4031 1.8184e-07 1.6195e-05D4 0.0237 0.1573 0.4262 1.6408e-07 3.1521e-03D3 0.0237 0.1795 0.4427 1.7470e-07 4.3539e-06D2 0.0185 0.1061 0.4059 6.6071e-08 1.7049e-06D1 0.0191 0.1186 0.4172 6.2552e-08 1.8425e-05T 0.0696 0.0925 0.0816 1.2325e-13 8.1328e-12

4.1.1 PCE

To build PCE of the MLE in terms of the 15 input random variables, weuse an experimental design comprising N = 1, 000 points drawn with Latinhypercube sampling (LHS) McKay et al. (1979). LHS is a popular techniquefor obtaining random experimental designs ensuring uniformity of each sam-ple on the margin input variables {X1, . . . , XM}. A histogram of the modelresponse at the considered LHS design is shown in Figure 6, indicating a pos-itively skewed distribution with the mode situated at MLE ≈ 75,000 years.

We develop two PCE meta-models, respectively denoted 1A and 1B, byapplying the procedure in Section 3.2.1, first, on the original scale and then,on the logarithmic transform of the MLE. For both PCE, the candidatebasis is determined using a hyperbolic truncation scheme (see Eq. (18)) withq = 0.5. The maximum degree p is varied from 1 to 15 and the optimal sparsePCE is selected by means of the corrected relative LOO error (see Eq. (23)),simply called LOO error and denoted errLOO hereafter.

19

0 0.05 0.1 0.15 0.2

T

D1

D2

D3

D4

C1

C2

C3ab

L1a

L1bL2aL2bL2c

K1K2

K3

φ

Figure 4: Porosity ranges along the model cross-section.

When the non-transformed response is considered, the optimal PCE 1Ais obtained for p = 10 and the corresponding LOO error is errLOO = 0.0321.The sparse meta-model includes 166 basis elements, whereas the total numberof basis elements for p = 10 and q = 0.5 (resp. q = 1) is 1, 656 (resp.3.2×106). When the logarithmic response is considered, the optimal PCE 1Bis obtained for p = 13 and the corresponding LOO error is errLOO = 0.0120.In this case, the sparse meta-model includes 245 basis elements, whereas thetotal number of basis elements for p = 13 and q = 0.5 (resp. q = 1) is 3, 801(resp. 3.7 × 107). For both PCE, the sparse bases involve polynomials inall 15 input variables. The index of sparsity, defined as the number of basiselements in the sparse expansion divided by the size of a full basis (q = 1)with the same maximum degree, is 166/3.2×106 ≈ 5.2×10−5 for case 1A and245/3.7× 107 ≈ 6.6× 10−6 for case 1B. These numbers indicate the interestin developing sparse PCE for such analyses.

The left graph of Figure 7 compares the values of PCE 1A, Y , with therespective values of the exact model, Y , at the input samples of the experi-mental design; the right graph shows a similar comparison for the exponentialtransform of PCE 1B. Assessing the relative accuracy of the two meta-models,first, we note that 1B yields a smaller LOO error; furthermore, it results in

20

10−12

10−10

10−8

10−6

10−4

10−2

T

D1

D2

D3

D4

C1

C2

C3ab

L1a

L1bL2aL2bL2c

K1K2

K3

Kx[m/s]

Figure 5: Hydraulic conductivity ranges along the model cross-section.

0 1 2 3 4 5

x 105

0

50

100

150

Fre

quency

MLE [years]

Figure 6: Histogram of mean lifetime expectancy values.

a smaller dispersion of Y −Y around zero and a better approximation of theexact response at the upper tail of its distribution. However, we should bearin mind that in the subsequent SA, the Sobol’ indices obtained from the coef-ficients of PCE 1B represent contributions to the variance of the logarithmicMLE.

21

0 1 2 3 4

x 105

0

1

2

3

4x 10

5 errLOO=0.032067

Y (x (i))

Y(x

(i) )

0 1 2 3 4

x 105

0

1

2

3

4x 10

5 errLOO=0.011952

Y (x (i))

Y(x

(i) )

Figure 7: Comparison of PCE 1A (left) and the exponential transform ofPCE 1B (right) with the actual model response at the experimental design.

4.1.2 Sobol’ indices

Figures 8 and 9 show bar-plots of the first-order and total Sobol’ indices ofthe porosities at the 15 layers using PCE 1A and 1B, respectively. To identifyunimportant effects, the threshold of 0.01 is marked with a horizontal dashedline.

Both figures indicate that the variability in the MLE is mainly due tomain effects and interactions associated with layers D4, L1b, C3ab and L1ain order of importance in terms of total effects, with layer D4 being clearlydominant. All aforementioned layers are located close to the host layer C2;D4 is the thickest among those and has the highest hydraulic conductivity.The condition SToti > 0.01 additionally classifies as important layers C1 andD1 for both PCE 1A and 1B, and marginally layers D3 and L2b for PCE 1Aonly. Note that although C1 is adjacent to the host layer C2, it is associatedwith smaller total and first-order indices than other neighboring layers, whichmay be attributed to its small thickness.

To gain further insight into the effects of the important variables on themodel response, we examine the behavior of first-order summands comprisingunivariate polynomials only, i.e.

M(1)i (xi) = E [M(X|Xi = xi)] (30)

or equivalently

M(1)i (xi) =

∑

α∈Ai

yαΨα(xi), Ai = {α ∈ A : αi > 0, αi 6=j = 0}. (31)

Figures 10 and 11 depict such univariate effects considering PCE 1A and1B, respectively, for the porosities at the six layers classified as important

22

K3 K1K2L2c L2b L2a L1b L1aC3ab C2 C1 D4 D3 D2 D1 T0.01

0.2

0.4

0.6

0.8Total Sobol’ Indices

STot

i

K3 K1K2 L2c L2b L2a L1b L1a C3ab C2 C1 D4 D3 D2 D1 T 0.01

0.2

0.4

0.6

0.8Sobol’ Indices Order 1

S(1)

i

Figure 8: Sobol’ indices using PCE 1A.

with both meta-models. The scales in the vertical axes in the two figuresare different due to consideration of the non-transformed response in oneand the logarithmic response in the other. The figures demonstrate that anincreasing porosity, and thus an increasing permeability, is associated with adecreasing algebraic contribution to the MLE value, though this relationshipis not strictly monotonic for all layers. Despite the different scales, the shapesof respective curves in the two figures demonstrate similar trends. However,we note the presence of higher-order terms for PCE 1B, which is consistentwith its higher degree p and lower sparsity as compared to 1A. For bothPCE, the sum in Eq. (31) tends to include more higher-order terms for theimportant variables.

For model 1A, main and second-order effects respectively account for88.5% and 10.3% of the response variance, indicating that the contributionof higher-order effects is merely 1.2%. For model 1B, main and second-ordereffects respectively account for 94.7% and 4.8% of the response variance,indicating that the contribution of higher-order effects is smaller than 1.0%.

23

K3 K1K2L2c L2b L2a L1b L1aC3ab C2 C1 D4 D3 D2 D1 T0.01

0.2

0.4

0.6

0.8Total Sobol’ Indices

STot

i

K3 K1K2 L2c L2b L2a L1b L1a C3ab C2 C1 D4 D3 D2 D1 T 0.01

0.2

0.4

0.6

0.8Sobol’ Indices Order 1

S(1)

i

Figure 9: Sobol’ indices using PCE 1B.

For both models, the highest second-order indices involve one of the layersD4 or L1b. Overall, effects from interactions are slightly more significant forthe variance of MLE in the original scale.

4.2 Case 2: 78 input random variables

In case 2, we employ a more detailed description of the input uncertainty. Inaddition to the petrofacies, the uncertain input in each layer comprises thefollowing four parameters governing the advective-dispersive processes: theanisotropy in the components of the hydraulic conductivity tensor, AK ; theEuler angle of the hydraulic conductivity tensor, θ; the longitudinal compo-nent of the dispersivity tensor, αL; and the anisotropy in the longitudinal andvertical components of the dispersivity tensor, Aα. As in case 1, a determin-istic relationship log10(Kx) = f(φ) is assumed for each layer. Furthermore,the uncertain input includes the hydraulic gradients, ∇H, at three zones:the Dogger sequence, the Oxfordian sequence and the top of the model,

24

0.05 0.1 0.15−5

0

5

x 104

φD4

M(1)

i

0.14 0.16 0.18−5

0

5

x 104

φL1b0.08 0.1 0.12

−5

0

5

x 104

φC3ab

0.1 0.15 0.2−5

0

5

x 104

φL1a

M(1)

i

0.02 0.04 0.06−5

0

5

x 104

φC10.05 0.1

−5

0

5

x 104

φD1

Figure 10: Univariate effects of important variables using PCE 1A.

0.05 0.1 0.15−0.4

−0.2

0

0.2

0.4

φD4

M(1)

i

0.14 0.16 0.18−0.4

−0.2

0

0.2

0.4

φL1b0.08 0.1 0.12

−0.4

−0.2

0

0.2

0.4

φC3ab

0.1 0.15 0.2−0.4

−0.2

0

0.2

0.4

φL1a

M(1)

i

0.02 0.04 0.06−0.4

−0.2

0

0.2

0.4

φC10.05 0.1

−0.4

−0.2

0

0.2

0.4

φD1

Figure 11: Univariate effects of important variables using PCE 1B.

respectively denoted zone 1, zone 2 and zone 3. In total, the model sub-sumes M = 78 independent input random variables upon the MLE of watermolecules outflowing from the TZ in the middle of layer C2.

The uncertain porosities, and the associated hydraulic conductivities, aremodeled as in case 1. Due to the lack of explicit information, each of theparameters AK , θ, αL and Aα is identically distributed in the different layers.In particular, the anisotropy ratios AK and Aα both follow a uniform distri-

25

bution in [0.01, 1], the Euler angle θ is uniformly distributed in [-30, 30] (indegrees) and the parameter αL is uniformly distributed in [5, 25] (in meters).The static hydraulic gradients are also uniformly distributed in the rangesgiven in Table 4. These were obtained by applying a perturbation of 20%of the nominal hydraulic gradient within each of the three zones. Althoughhypothetical, these conservative uncertainty ranges were purposely chosen toprovide insights into the behavior of the multi-layered model.

Table 4: Ranges of hydraulic gradient at the three zones of interest.

Zone number ∇H(min) ∇H(max)

1 0.00064 0.000962 0.00240 0.003603 0.00272 0.00408

4.2.1 PCE

In this case, the available data for building PCE consist of the MLE values for(i) an experimental design of size N = 2, 000, drawn with LHS and denotedE , and (ii) an enrichment of E of equal size N ′ = 2, 000, denoted E ′. Theenrichment is built so that the joint set {E , E ′} forms a LHS experimentaldesign as well. Histograms of the model response for the two sets of inputvectors are shown in Figure 12. Positively skewed distributions are observedfor both output sets, while modes are situated at MLE ≈ 85,000 years.

0 1 2 3 4 5

x 105

0

50

100

150

200

250

Fre

quency

MLE [years]0 1 2 3 4 5

x 105

0

50

100

150

200

250

Fre

quency

MLE [years]

Figure 12: Histograms of mean lifetime expectancy values calculated withsets E (left) and E ′ (right).

In the following, we develop PCE based on E and the joint set {E , E ′}.For the set E , we consider the MLE response in both the original and the

26

logarithmic scales; in this case, the enrichment E ′ serves as a validation setfor computing the generalization error (see Section 3.2.2). For all PCE, thecandidate basis is determined using a hyperbolic truncation scheme withq = 0.5. The maximum degree p is varied from 1 to 15 and the optimalsparse meta-model is selected by means of the LOO error.

The first PCE, denoted 2A, is built using E as the experimental designand considering the response in the original scale. The optimal degree isp = 8 and the corresponding LOO error is errLOO = 0.0565. The sparse PCEincludes 185 basis elements, whereas the total number of basis elements forp = 8 and q = 0.5 (resp. q = 1) is 18, 643 (resp. 5.3 × 1010). Thus, theindex of sparsity is 185/5.3 × 1010 ≈ 3.5 × 10−9. The sparse basis consistsof polynomials in 68 out of the 78 total input parameters, meaning that theoutput does not depend at all on the values of the 10 excluded parameters.Note that 3 out of the 10 excluded parameters are properties of layer T. Thegeneralization error (evaluated with E ′) is errG = 0.0759. The left graphof Figure 13 compares the values of the meta-model, Y , with the respectivevalues of the exact model, Y , at the input samples of the experimental design,E . A similar comparison but for the validation set, E ′, is shown in the rightgraph of the same figure.

0 1 2 3 4 5

x 105

0

1

2

3

4

5x 10

5 errLOO=0.056488

Y (x (i))

Y(x

(i) )

0 1 2 3 4 5

x 105

0

1

2

3

4

5x 10

5 errG=0.07591

Y (x i)

Y(x

i)

Figure 13: Comparison of PCE 2A with the actual model response at theexperimental design, E , (left) and at the validation set, E ′ (right).

A second PCE, denoted 2B, is built by using again E as the experimentaldesign, but employing a logarithmic transform of the MLE. The optimalsparse PCE is obtained for p = 8 (same degree as for 2A) and comprises 163basis elements; the corresponding LOO error is errLOO = 0.0287. The sparsebasis consists of polynomials in 65 out of the 78 total input parameters;note that 6 out of the 13 excluded parameters are dispersivity anisotropyratios (Aα). The resulting generalization error for the MLE response in the

27

original scale (considering the exponential transform of the obtained PCE)is errG = 0.0452, which is slightly lower than that of PCE 2A. The left andright graphs of Figure 14 compare the exponential transform of PCE 2B withthe exact model response at E and E ′, respectively.

0 1 2 3 4 5

x 105

0

1

2

3

4

5x 10

5 errLOO=0.02866

Y (x (i))

Y(x

(i) )

0 1 2 3 4 5

x 105

0

1

2

3

4

5x 10

5 errG=0.045202

Y (x i)

Y(x

i)

Figure 14: Comparison of exponential transform of PCE 2B with the actualmodel response at the experimental design, E , (left) and at the validationset, E ′ (right).

Finally, we use as experimental design the joint set {E , E ′} consisting ofN + N ′ = 4, 000 points. The optimal PCE is obtained for p = 10 and andthe corresponding LOO error is errLOO = 0.0384. The sparse PCE comprises312 basis elements and has an index of sparsity 312/4.5×1012 ≈ 6.9×10−11.The only two parameters excluded from the sparse basis are AL2bK and αTL.The comparison between the meta-model, denoted 2C, and the exact modelresponse at the input samples of the experimental design, {E , E ′}, is shownin Figure 15.

Assessing the relative accuracies of the three meta-models, we note thatall have LOO errors of the same order of magnitude, with the smallest errorcorresponding to PCE 1B. Because it is of interest to limit the number ofcostly evaluations of the exact hydrogeological model, an experimental designcomprising 2, 000 points is deemed most appropriate. We therefore conductSA for the MLE response by post-processing the coefficients of PCE 2A and2B and compare the results. We should bear in mind that the Sobol’ indicesobtained from the coefficients of 2B represent contributions to the varianceof the logarithmic MLE.

28

0 1 2 3 4 5

x 105

0

1

2

3

4

5x 10

5 errLOO=0.038411

Y (x(i))

Y(x

(i) )

Figure 15: Comparison of PCE 2C with the actual model response at theexperimental design, {E , E ′}.

4.2.2 Sobol’ indices

Figures 16 and 17 show bar-plots of the total and first-order Sobol’ indicesfor PCE 2A and 2B, respectively. In each graph, bar-plots of the ten largestindices are presented in descending order. The superscripts on the parametersymbols on the horizontal axes denote layer names or zone numbers. Toidentify unimportant effects, the threshold of 0.01 is marked with a horizontaldashed line.

A first observation is that both PCE lead to the same ranking of the fivevariables with the highest total indices. These are the porosities of layers D4,C3ab, L1b, L1a and C1 in order of importance, with the porosities of layer D4being the by-far dominant. For both PCE, these are also the variables withthe highest first-order indices following the same ranking. Note that PCE1A and 1B identified the same five layers as most significant in terms of bothtotal and main effects, which indicates the consistency of the analyses in cases1 and 2. However, the ranking of layers C3ab and L1b in terms of the totalindices is interchanged in the two cases; also, layers D4 and C1 have slightlyhigher contributions in case 2. Employing the criterion SToti < 0.01 to sortout unimportant variables, the porosities of the aforementioned five layerscomprise the only important parameters for both 2A and 2B. Univariateeffects for these variables follow similar trends as in case 1 and are thusnot shown. The screening allows one to consider 73 out of 78 parameters asunimportant, meaning that they could be given a deterministic value withoutaffecting essentially the predicted MLE.

Let us now examine higher-order effects. For both PCE, the largestsecond-order effects comprise interactions between the five porosities clas-

29

0.01

0.2

0.4

0.6

0.8

φD4 φC3ab φL1b φL1a φC1 φD1 ∇H1 φL2b φL2aaC3ab

L

Total Sobol’ IndicesSTot

i

0.01

0.2

0.4

0.6

0.8

φD4 φC3ab φL1b φL1a φC1 aC3abL

AD4K

∇H1 φD1 φC2

Sobol’ Indices Order 1

S(1)

i

Figure 16: Sobol’ indices using PCE 2A.

sified above as important and involve one of the layers D4 or L1b; theirsum explains 10.9% and 4.2% of the total variance for PCE 2A and 2B, re-spectively. Overall, second-order effects are slightly less important for thelogarithmic response than for the response in the original scale, which is con-sistent with the results in case 1. Contributions of higher than second-ordereffects are practically zero.

The above analysis indicated that among the set of random variables,only the porosities of certain layers are important for explaining the responsevariance. As highlighted earlier, because of the assumed relationship betweenporosity and hydraulic conductivity in each layer, the Sobol’ indices for the φvariables are also indicative of the importance of Kx in the respective layers.Accordingly, in the subsequent discussion of the SA results (Section 4.3), wewill interpret the variability of MLE in terms of joint effects of the petrofaciesP .

For a more in-depth investigation of the contributions of the differenttypes of hydro-dispersive parameters, Table 5 lists the sums of the first-order

30

0.01

0.2

0.4

0.6

0.8

φD4 φC3ab φL1b φL1a φC1 AD4

KaC3abL

φD1 φL2a ∇H1

Total Sobol’ IndicesSTot

i

0.01

0.2

0.4

0.6

0.8

φD4 φC3ab φL1b φL1a φC1 aC3abL

AD4K

∇H2∇H1 φD1

Sobol’ Indices Order 1

S(1)

i

Figure 17: Sobol’ indices using PCE 2B.

indices per type of property over all layers or zones of the considered cross-section. According to this table, the added main effects of the porositiesaccount for approximately 87% − 93% of the response variance for the twoPCE, whereas the added main effects of all remaining input random variablesaccount for a mere 2.5% − 2.8%. We note the slightly higher contributionsof main effects for 2B and the zero main effects of Aα for both PCE.

Table 5: Sums of first-order Sobol’ indices over all layers per type of property.

meta-model φ AK θ αL Aα ∇H2A 0.8664 0.0088 0.0029 0.0076 0 0.00572B 0.9302 0.0096 0.0032 0.0088 0 0.0061

31

4.3 Discussion of results

In both cases 1 and 2, the petrofacies of aquifer formations have the largesteffects on the variability of the MLE at the TZ. For layers D4 and L1b inparticular, for which the upper bounds of the hydraulic conductivity rangesare the highest (see Table 3, Figure 5), strong advective processes within theirown volume may reduce the overall groundwater residence time in the model.Besides, the wider the ranges of Kx values in these permeable formations, thehigher are the contributions of the respective petrofacies to the variability ofthe MLE. The position of the layers relatively to layer C2 is also relevant:the further the layer is, the lower is its effect on the MLE of water moleculesdeparting from the TZ. Layer L1a, which is the first aquifer encounteredin the Oxfordian sequence (at a distance of 60 meters from layer C2), hasa significant effect on the output variance, whereas layers D2 and D3 thathave similar Kx ranges have much smaller contributions.

The three most significant aquifer formations, namely D4, L1b and L1a,have rather non-linear univariate effects on the output response (see Figures10 and 11). Substantial changes within the response are observed with smallshifts in the ranges of low porosity-permeability values, revealing the effect ofthe balance between advective and dispersive-diffusive transport processes.But for higher porosity-permeability values, advective fluxes prevail, thusyielding more linear and moderate changes in the response.

The relatively large uncertainty range and high upper-bound value of per-meability for layer D1 results in a marginal effect on the response variabilityin case 1 (see Figures 8 and 9). The uncertainty in the petrofacies of thislayer influences linearly or nearly linearly and rather moderately the transittime of water molecules from the TZ (see Figures 10 and 11). However, ac-counting for uncertain dispersion processes and groundwater fluxes in case 2may reduce the quantity of water molecules departing from the TZ and reach-ing layer D1, thus lessening the contribution of PD1 to the variance of theoutput quantity, which becomes unimportant with respect to the thresholdSToti < 0.01.

The petrofacies of semi-permeable formations, PC3ab and PC1, are alsosignificantly influencing the variability of the MLE at the TZ. By isolatinglayer C2 from major aquifer formations, they buffer solute intrusions into theOxfordian and Dogger aquifer sequences, thus acting like a geological barrier.Figures 10 and 11 show that their univariate effects on the output quantityare relatively non-linear despite their limited amplitude.

We underline that the petrofacies of the host layer attributed to theCallovo-Oxfordian claystone (PC2) are insignificant with regard to the MLEvariability. Slow diffusive processes take place into highly impermeable rocks,

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which induces large values of the MLE response. Modifying the magnitudesof advective-dispersive transport processes in layer C2 does not add a signif-icant variability to the time required for solutes to leave the layer’s volume.

The magnitude of the transverse advective fluxes in each layer is related tothe respective value of Kz, in which the uncertainty is accounted for throughthe anisotropy ratio AK . Although AK represents the second most influentialgroup of factors, considering added effects from all layers (see Table 5), theuncertainty in this property adds a small amount of variability to the MLE(< 1%) compared to the petrofacies (≈ 90%). We note however that fac-tor AD4

K is only marginally excluded from the important factors when PCE

2B is considered (see Figure 17). Indeed, for the highest Kx values, strongadvective fluxes take place within the layer’s volume. Under the assump-tion of strong transverse fluxes (AD4

K → 1), solutes can be oriented towardneighbouring layers where slower fluxes occur, thus raising the MLE.

For each layer, the Euler angle θ could deviate groundwater fluxes from anorientation parallel to the x-axis and toward the main discharge boundaries,thus raising the variability of the response. Although it could be assumedas especially influential in the most advective layers, the total contributionof this group of uncertain factors to the variance of the MLE is negligible incomparison to that of the petrofacies P (see Table 5).

In aquifer formations, the effects of the uncertainty regarding the macro-dispersion tensors upon the response quantity, i.e. the magnitude (αL) andanisotropy (Aα) of the tensors, are concealed by the strong effect of petrofa-cies on the advective part of the transport processes (see Table 5). We notehowever that the longitudinal component of the macro-dispersion tensor inlayer C3ab (αC3ab

L ) appears among the ten factors with the highest totalSobol’ indices for both PCE 2A and 2B. The anisotropy ratios, Aα, have nocontribution at all to the response variance when considered independently;the uncertainty in these factors contributes to the variability of MLE onlythrough interaction terms.

The sensitivity of the MLE with respect to flow BCs considered in themodel is directly related to the magnitude and orientation of the advectivefluxes in the entire model. In the case of high gradients in both limestonesequences, the advective solute transport processes would raise within theirvolume, and thus reduce the MLE. Table 5 indicates that the three randomhydraulic gradients have a small added effect on the output variance. Notehowever that the total Sobol’ index for the hydraulic gradient in the Doggersequence (∇H1) belongs to the ten highest indices for both PCE 2A and2B. The uncertainty regarding this factor can alter the advective processesoccurring notably in layer D4, which has the far highest contribution to the

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variance of the MLE calculated at the TZ.

5 Conclusions

The model introduced in this paper stands as a two-dimensional verticalcross-section of the subsurface of Paris Basin in the vicinity of Bure (Haute-Marne). While encompassing most of the hydrogeological features of theunderground media, it was simplified with regard to geometries, disconti-nuities, fractures and heterogeneities. This numerical model is intended toexplore the behavior of a complex multi-layered hydrogeological system atlow computational cost and provide insights into the effect of uncertain pa-rameters upon various types of model output.

Sensitivity analysis (SA) was carried out for two cases of modeling theinput uncertainty. For the sake of simplicity, homogeneous parameters wereassumed within each of the 15 hydrogeological layers comprising the modelin both cases. In case 1, only the petrofacies, P , regarded as the couplepermeability-porosity, {K,φ}, were considered uncertain. In case 2, the un-certain factors at each layer included four additional hydro-dispersive param-eters: the anisotropy ratio and the orientation of the hydraulic conductivitytensor, AK and θ respectively, the magnitude and anisotropy ratio in themacro-dispersion tensor, αL and Aα respectively; additionally, the hydraulicgradients, ∇H, in three zones of the numerical model were considered ran-dom, summing up to 78 uncertain input factors.

In the present study, a target zone (TZ) located within the middle layer(C2) of the model provides the output response of interest. Latin hyper-cube sampling was employed to address the propagation of the uncertaintyfrom the input factors upon the mean lifetime expectancy (MLE) of watermolecules departing from the TZ. Polynomial chaos expansion (PCE) meta-models were used to compute the Sobol’ sensitivity indices for each inputfactor at low computational costs. Sparse PCE proved particularly efficientin providing fairly accurate representations of the MLE in terms of the con-sidered high-dimensional input. The accuracy was enhanced when the PCEwere fitted to the logarithmic MLE.

Focusing on the effects of petrofacies solely, case 1 demonstrated the largecontributions of aquifer formations to the variance of the model output. Inparticular, (i) the closer the aquifer formation to layer C2, (ii) the thickerthe layer, (iii) the wider the ranges of the petrofacies and (iv) the higherthe upper bound of the hydraulic conductivity range, the larger were theeffects of the petrofacies on the variability of the response. Investigationof the univariate effects of petrofacies highlighted that for these permeable

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formations, the response is more sensitive to changes occurring within lowporosity-permeability ranges. Hence, within a certain range of {K,φ} values,the dispersive-diffusive processes counterbalance with the strong advectivefluxes in the ageing process.

The SA results in case 2 showed that the variability of the MLE is almostentirely due to the uncertainty regarding the petrofacies of the hydrogeo-logical layers. The other hydro-dispersive parameters are insignificant forexplaining the response variance and may be considered as deterministic fac-tors in future works. SA in cases 1 and 2 were consistent in identifying thelayers of which the petrofacies are important.

In formations characterized by highly advective processes, the longitudi-nal hydraulic conductivities applying in the main groundwater direction havelarge contributions to the MLE variability. The two semi-confining forma-tions encompassing the C2 layer buffer the transfer of solute from the lattertoward the further aquifer sequences. Besides, it is acknowledged that lon-gitudinal dispersion processes occurring within their own volume also retardthe solute transfer toward the adjacent aquifers. Because of the diffusion-dominated transport processes occurring within its volume, the petrofaciesof the highly-confining C2 layer have a negligible effect on the variance ofthe output response.

It is important to remind that the use of a 2D model tends to overestimatethe output response of interest by omitting the advection and dispersionalong the third dimension. Recognizing this limitation, we underline thatthe purpose of the synthetic model introduced in this study is to shed lighton the relative effects of various uncertain factors governing the advectiveand dispersive processes in a complex multi-layered hydrogeological system.The methods employed in this study can be applied to a real-case studyemploying a realistic 3D numerical model.

The sensitivity analysis performed in this work is deemed very informa-tive for future applications with the high-resolution integrated Meuse/Haute-Marne hydrogeological model. In the frame of a real-case uncertainty analysiswith concern to a solute transport in the subsurface of the Paris Basin, theauthors recommend defining as thoroughly as possible the spatial distribu-tions of hydraulic conductivity values, with a main focus on the large aquifersequences of Oxfordian and Dogger ages.

AcknowledgementThe authors are very grateful to Mr Benjamin Brigaud (Universite Paris-

Sud) and Mrs Agnes Vinsot (ANDRA) for having provided the hydro-dispersivedatasets used in this analysis, and to Mr Fabien Cornaton (DHI-WASYGmbH) for providing the code GroundWater.

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